Dual Steenrod algebra

In algebraic topology, through an algebraic operation (dualization), there is an associated commutative algebra[1] from the noncommutative Steenrod algebras called the dual Steenrod algebra. This dual algebra has a number of surprising benefits, such as being commutative and provided technical tools for computing the Adams spectral sequence in many cases (such as [2]pg 61-62) with much ease.

Definition

Recall[2]pg 59 that the Steenrod algebra (also denoted ) is a graded noncommutative Hopf algebra which is cocommutative, meaning its comultiplication is cocommutative. This implies if we take the dual Hopf algebra, denoted , or just , then this gives a graded-commutative algebra which has a noncommutative comultiplication. We can summarize this duality through dualizing a commutative diagram of the Steenrod's Hopf algebra structure:

If we dualize we get maps

giving the main structure maps for the dual Hopf algebra. It turns out there's a nice structure theorem for the dual Hopf algebra, separated by whether the prime is or odd.

Case of p=2

In this case, the dual Steenrod algebra is a graded commutative polynomial algebra where the degree . Then, the coproduct map is given by

sending

where .

General case of p > 2

For all other prime numbers, the dual Steenrod algebra is slightly more complex and involves a graded-commutative exterior algebra in addition to a graded-commutative polynomial algebra. If we let denote an exterior algebra over with generators and , then the dual Steenrod algebra has the presentation

where

In addition, it has the comultiplication defined by

where again .

Rest of Hopf algebra structure in both cases

The rest of the Hopf algebra structures can be described exactly the same in both cases. There is both a unit map and counit map

which are both isomorphisms in degree : these come from the original Steenrod algebra. In addition, there is also a conjugation map defined recursively by the equations

In addition, we will denote as the kernel of the counit map which is isomorphic to in degrees .

See also

References

  1. Milnor, John (2012-03-29), "The Steenrod algebra and its dual", Topological Library, Series on Knots and Everything, WORLD SCIENTIFIC, Volume 50, pp. 357–382, doi:10.1142/9789814401319_0006, ISBN 978-981-4401-30-2, retrieved 2021-01-05
  2. Ravenel, Douglas C. (1986). Complex cobordism and stable homotopy groups of spheres. Orlando: Academic Press. ISBN 978-0-08-087440-1. OCLC 316566772.
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